Summary: TOWARDS OPTIMAL SCALING OF METROPOLIS-COUPLED
MARKOV CHAIN MONTE CARLO
YVES F. ATCHAD´E, GARETH O. ROBERTS, AND JEFFREY S. ROSENTHAL
Abstract. We consider optimal temperature spacings for Metropolis-coupled
Markov chain Monte Carlo (MCMCMC) and Simulated Tempering algorithms.
We prove that, under certain conditions, it is optimal (in terms of maximising
the expected squared jumping distance) to space the temperatures so that the
proportion of temperature swaps which are accepted is approximately 0.234. This
generalises related work by physicists, and is consistent with previous work about
optimal scaling of random-walk Metropolis algorithms.
The Metropolis-coupled Markov chain Monte Carlo (MCMCMC) algorithm ,
also known as parallel tempering or the replica exchange method, is a version of
the Metropolis-Hastings algorithm [17, 9] which is very effective in dealing with
problems of multi-modality. The algorithm works by simulating multiple copies
of the target (stationary) distribution, each at a different temperature. Through a
swap move, the algorithm allows copies at lower temperatures to borrow information
from high-temperature copies and thus escape from local modes. The performance
of MCMCMC depends crucially on the temperatures used for the different copies.
There is an ongoing discussion (especially in the Physics literature) on how to best